answer:To solve this problem, let's start by understanding the given equation and leveraging geometric properties of complex numbers.The equation ( |z + mathrm{i}| + |z - mathrm{i}| = 2 ) describes the set of points whose sum of distances to the points ( mathrm{i} ) and ( -mathrm{i} ) is constant and equal to 2. This can be recognized as the definition of an ellipse with foci at ( mathrm{i} ) and ( -mathrm{i} ) and the length of the major axis equal to 2.However, since the distance (major axis) is exactly the sum of the distances between the points (the foci), this is actually reducing our ellipse to the line segment between (mathrm{i}) and (-mathrm{i}). Therefore, the only solutions for such ( z ) are points on the line segment joining ( mathrm{i} ) and ( -mathrm{i} ).This means ( z = xmathrm{i} ) for ( x ) between -1 and 1. Let's denote this general point as ( z = mathrm{i} y ), where ( y ) is a real number such that (-1 le y le 1 ).We are asked to find the minimum value of ( |z + mathrm{i} + 1| = |mathrm{i} y + mathrm{i} + 1| = |1 + mathrm{i}(y + 1)| ).The problem now reduces to finding the minimum value of:[ |1 + mathrm{i}(y + 1)| ]over the interval (-1 le y le 1).Let's calculate this using Python.```pythonimport numpy as np# Define the function to calculate the magnitudedef magnitude(y): return np.abs(1 + 1j * (y + 1))# Generate y values from -1 to 1y_values = np.linspace(-1, 1, 500)# Evaluate the magnitudes for these y valuesmagnitudes = [magnitude(y) for y in y_values]# Find the minimum magnitudemin_magnitude = min(magnitudes)print(min_magnitude)``````output1.0```The minimum value of ( |1 + mathrm{i}(y + 1)| ) over the interval (-1 le y le 1) is boxed{1}.Thus, the final answer is ( A. 1 ).

answer:To determine which of the four numbers is the product of the integers from 1 to 15 (also known as 15!), we can compare each of these numbers to the factorial of 15. We'll first compute 15!, then compare each of the given numbers with 15! to find the match:[15! = 1 times 2 times 3 times dots times 15]The given numbers are:1. 13076743680002. 13076743685003. 13076743682004. 1307674368010Let's compute 15! and check each number.```pythonimport math# Compute 15!factorial_15 = math.factorial(15)# Given numbersnumbers = [1307674368000, 1307674368500, 1307674368200, 1307674368010]# Check which number among the given matches 15!result = [number for number in numbers if number == factorial_15]print(result)``````output[1307674368000]```The number (1307674368000) is the product of the integers from 1 to 15.Thus, the answer is (boxed{1307674368000}).

answer:"Les Miserables" is a complex work that explores several themes. Some of the primary ones include the struggle between good and evil, the importance of love and compassion, the critique of social injustice, and the power of redemption and change. The narrative demonstrates these themes through the life experiences of its characters, particularly its protagonist, Jean Valjean. Please note that this is a simplified summary, and the novel delves into these themes in a much deeper and nuanced way.

answer:To solve this problem, we can enumerate all possible ways to paint the (2 times 2) grid where no green square shares its top or right side with any red square. We will label the squares as follows:[begin{array}{|c|c|}hlineA & B hlineC & D hlineend{array}]We need to ensure that:1. If (A) is green, then (B) and (C) cannot be red.2. If (B) is green, then (D) cannot be red.3. If (C) is green, then (D) cannot be red.Let's enumerate all valid configurations using Python and sympy.```pythonfrom sympy import symbols, Eq, Ordef is_valid_configuration(A, B, C, D): """ Checks if the given configuration is valid. A, B, C, D are either 0 (red) or 1 (green). """ if (A == 1 and (B == 0 or C == 0)): return False if (B == 1 and D == 0): return False if (C == 1 and D == 0): return False return True# Generate all possible configurationsconfigurations = [ (A, B, C, D) for A in [0, 1] for B in [0, 1] for C in [0, 1] for D in [0, 1]]# Filter valid configurationsvalid_configurations = [ (A, B, C, D) for A, B, C, D in configurations if is_valid_configuration(A, B, C, D)]# Number of valid configurationsnum_valid_configurations = len(valid_configurations)print(num_valid_configurations)``````output6```The number of different ways to paint the squares so that no green square shares its top or right side with any red square is (boxed{6}).

answer:To solve the problem, we need to determine the number of real roots of the equation:[frac{left(x^{2}-x+1right)^{3}}{x^{2}(x-1)^{2}}=frac{left(pi^{2}-pi+1right)^{3}}{pi^{2}(pi-1)^{2}}]Let's break it down into manageable steps:1. Understand the Equation: The equation is of the form (frac{f(x)}{g(x)} = c) where (c) is a constant. Here, (c) is given by (frac{left(pi^{2}-pi+1right)^{3}}{pi^{2}(pi-1)^{2}}).2. Simplify the Problem: We need to find all values of (x) for which: [ frac{left(x^{2}-x+1right)^{3}}{x^{2}(x-1)^{2}} = text{constant} ] This can be rewritten as: [ left(x^{2}-x+1right)^{3} = c cdot x^{2}(x-1)^{2} ]3. Solve the Equation: We need to find the roots of the equation after substituting the constant (c).Let's implement this in Python using SymPy to find the number of real roots.```pythonimport sympy as sp# Define the variablex = sp.symbols('x')# Define the left-hand side and the constant termlhs = (x2 - x + 1)3 / (x2 * (x - 1)2)constant = (sp.pi2 - sp.pi + 1)3 / (sp.pi2 * (sp.pi - 1)2)# Define the equationequation = sp.Eq(lhs, constant)# Solve for xsolutions = sp.solve(equation, x)# Filter real solutionsreal_solutions = [sol.evalf() for sol in solutions if sol.is_real]num_real_solutions = len(real_solutions)print(num_real_solutions)``````output6```The number of real roots of the equation [frac{left(x^{2}-x+1right)^{3}}{x^{2}(x-1)^{2}}=frac{left(pi^{2}-pi+1right)^{3}}{pi^{2}(pi-1)^{2}}]is (boxed{6}).This corresponds to option D.

answer:To solve the problem, we need to find the minimum value of the expression (frac{y}{x} + frac{1}{y}) given the constraints (x > 0), (y > 0), and (x + 2y = 1).Here's the step-by-step approach:1. Understand the Constraint: The constraint given is (x + 2y = 1).2. Express (x) in terms of (y): From the constraint, we can express (x) as (x = 1 - 2y).3. Substitute (x) into the expression: Substitute (x = 1 - 2y) into the expression (frac{y}{x} + frac{1}{y}).4. Simplify the Expression: Simplify the resulting expression and find its critical points to determine the minimum value.5. Verify the Constraint: Ensure that the critical points satisfy the constraint (x > 0) and (y > 0).Let's implement this in Python using SymPy to find the minimum value of the expression under the given constraints.```pythonimport sympy as sp# Define variablesx, y = sp.symbols('x y')# Given constraint: x + 2y = 1constraint = sp.Eq(x + 2*y, 1)# Solve the constraint for xx_expr = sp.solve(constraint, x)[0]# Define the expression to minimize: y/x + 1/yexpression = y/x + 1/y# Substitute x in the expressionexpression_substituted = expression.subs(x, x_expr)# Simplify the expressionexpression_simplified = sp.simplify(expression_substituted)# Find the critical points by taking the derivative with respect to y and setting it to 0critical_points = sp.solve(sp.diff(expression_simplified, y), y)# Evaluate the expression at the critical pointsmin_value = float('inf')for point in critical_points: if point > 0: value = expression_simplified.subs(y, point) if value < min_value: min_value = valueprint(min_value)``````output0```It seems there might be an issue with the solution or the evaluation of the critical points. Let’s revisit the problem and ensure we correctly handle the calculations and constraints.Given ( x + 2y = 1 ), we express ( x ) as ( x = 1 - 2y ). Substituting this into the expression:[frac{y}{x} + frac{1}{y} = frac{y}{1 - 2y} + frac{1}{y}]We need to minimize this expression with respect to ( y ), subject to ( 0 < y < frac{1}{2} ) (since ( x > 0 ) implies ( y < frac{1}{2} )).Let's re-implement the code with careful handling of the domain constraints for ( y ):1. Substitute ( x = 1 - 2y ) into the expression.2. Find the derivative of the substituted expression with respect to ( y ).3. Solve for critical points within the valid range of ( y ).4. Evaluate the expression at the critical points to find the minimum value.```pythonimport sympy as sp# Define variablesy = sp.symbols('y')# Express x in terms of yx_expr = 1 - 2*y# Define the expression to minimize: y/(1 - 2*y) + 1/yexpression = y / x_expr + 1 / y# Simplify the expressionexpression_simplified = sp.simplify(expression)# Find the critical points by taking the derivative with respect to y and setting it to 0derivative = sp.diff(expression_simplified, y)critical_points = sp.solve(derivative, y)# Evaluate the expression at the critical points within the valid range (0 < y < 1/2)min_value = float('inf')for point in critical_points: if 0 < point < 1/2: value = expression_simplified.subs(y, point) if value < min_value: min_value = value# Also check the endpoints of the valid rangeendpoints = [0, 1/2]for endpoint in endpoints: if 0 < endpoint < 1/2: value = expression_simplified.subs(y, endpoint) if value < min_value: min_value = valueprint(min_value)``````output4```The minimum value of the expression (frac{y}{x} + frac{1}{y}) given the constraints (x > 0), (y > 0), and (x + 2y = 1) is:[boxed{4}]This value is obtained by correctly substituting (x = 1 - 2y) into the expression, simplifying it, and finding the critical points within the valid range for (y).The minimum value of the expression (frac{y}{x} + frac{1}{y}) given the constraints (x > 0), (y > 0), and (x + 2y = 1) is (boxed{4}).